Block Numerical Range and Estimable Decomposition of Some Hyponormal Operators

In this paper we investigate the block numerical range and the existences of estimable decompositions of bounded linear operators on a separable Hilbert space. By using spectral measure, we show that there exists an estimable decomposition for the spectrum of every bounded normal operator. Furthermore, the corresponding result also holds for hyponormal operators with totally disconnected spectra. Finally, we obtain that for spectral operator, there exists an estimable decomposition, under quasi-nilpotent equivalence.